PRECURSORS OF MODERN LOGIC: BOLZANO

The most important logician of the first half of the nineteenth century was Bernard Bolzano (1781–1848). His views are closest to those of Leibniz, who preceded him by more than a century (Bolzano was sometimes called the Bohemian Leibniz). Although he quoted often and extensively from philosophers and logicians of his own generation and the preceding one, among them Kant, Salomon Maimon, Hegel, J. F. Fries, J. G. E. Maass, and K. L. Reinhold, he did this almost always in order to criticize them, and rightly so from our modern point of view, because orders of magnitude separate Bolzano as a logician from his contemporaries.

One may doubt whether he deserves to be called a forerunner of mathematical logic and modern semantics. His approach is in many respects rather crude and old-fashioned in comparison with those of George Boole and Gottlob Frege, one and two generations later, respectively. But many points first made by Bolzano look strikingly modern. Unfortunately most of these were either not noticed or not understood during his lifetime or were forgotten by later generations.

For Bolzano logic was mainly the theory of science. To investigate science he used a partly formalized language consisting of ordinary German extended by various types of constants and variables, as well as by certain technical terms which for the most part he was at great pains to define as carefully as possible.

The fundamental entities with which logic has to deal, according to Bolzano, are terms and the propositions they constitute. These abstract entities are carefully distinguished from the corresponding linguistic and mental entities. Because a single proposition can be expressed in an indefinite number of ways, Bolzano's first aim was to normalize such linguistic expressions, to reduce all of them to canonical forms prior to their purely formal treatment.

Bolzano's solution was highly idiosyncratic. Deviating radically from tradition, he claimed that all sentences (complex and compound sentences as well as simple ones) are reducible to the single form "A has b," where "A " is the subject term, "b " the predicate term, and "has" the copula. Although this reduction works reasonably well with such sentences as "John is hungry," which can easily be rendered as "John has hunger," it sounds less convincing in the case of reducing "This is gold" to "This has goldness" (although Bolzano presented reasons why such words as "goldness" had not been created in natural languages) and still less so when "John is not hungry" is reduced to "John has lack-of-hunger." The reduction of the compound sentence "Either P or Q " to "The-term-One-of-P -and-Q -is-true has the-property-of-being-a-singular-term" or "The-term-One-of-P -or-Q -is-true has nonemptiness" (depending on whether the original expression "Either … or …" is interpreted from its context as denoting exclusive or inclusive disjunction) looks rather strange in its verbal formulation, although it looks much less strange in some appropriate symbolism. And reducing "Some A is B " to "The-term-An-A -which-is-B has nonemptiness" may appear fantastic at first sight, although it looks much more familiar when symbolized as A ∩ B ≠ 0. Nevertheless, Bolzano did not attempt to present a full set of rules for such conversions and relied instead on the reader's willingness to believe in the existence of such reductions after being shown how to perform them on certain representative samples, including some rather recalcitrant cases.

This reduction played a small role in the further development of Bolzano's work in logic. His major innovation was his introduction of the technique of variation into what amounts essentially to the logical semantics of language, even though the semantic approach, in its modern sense, was foreign to him. Starting with a proposition, true or false, he investigated its behavior with regard to truth and falsehood under substitution for any of its terms of all other fitting (that is, propositionhood-preserving) terms. (In modern terminology, he investigated all models of sentential forms.) When the number of such variants was finite he defined the degree of validity of a proposition with respect to one or more of its constituent terms as the ratio of the number of its true variants to the number of all variants. When this ratio is 1, the proposition is universally valid; when 0, universally contravalid; when greater than 0, consistent.

After extending these notions to propositional classes Bolzano was able to define an amazing number of interesting, and sometimes highly original, metalogical notions, including compatibility, dependency, exclusion, contradictoriness, contrariety, exclusiveness, and disjointness. By far the most important notion introduced in this way is that of derivability with respect to a given class of terms, defined as holding between two propositions P and Q if and only if Q is consistent and every model of Q is a model of P with respect to this class of terms; with respect to propositional classes it is defined similarly. This definition differs only in the unfortunate consistency clause from Tarski's definition, given in 1937, of what he called the consequence relation.

Kant had defined an "analytic" affirmative judgment as one in which the predicate concept was already contained in the subject concept. Rejecting this definition as clearly inadequate for explicating logical truth, Bolzano defined a proposition to be analytically true when universally valid with respect to at least one of its constituent terms, analytically false when universally contravalid, etc., and as analytic when either analytically true or analytically false. Bolzano was aware that this definition of analytical truth was too broad as an explication of logical truth, and he therefore went on to define a proposition as being logically analytic when (again in modern terminology) all its descriptive (extralogical) constituent terms occur in it vacuously, an anticipation of a well-known definition by W. V. Quine (1940).

Bolzano's views of probability are also strikingly modern. To define the probability of the proposition M on the assumptions A, B, C, D, · · · (with respect to certain terms i, j, · · ·) he used the relative degree of validity of M with respect to A, B, C, D, · · ·, which he defined as the ratio of the number of true variants of the set M, A, B, C, D, · · · to the number of true variants of the set A, B, C, D, · · ·. This conception, tenable, of course, only when the numbers involved are finite, is an important refinement of Laplace's well-known conception of probability, standard in Bolzano's time, in that it elegantly sidesteps the problem of circularity involved in the notion of equipossibility.

Bibliography

Bar-Hillel, Yehoshua. "Bolzano's Propositional Logic." Archiv für mathematische Logik und Grundlagenforschung 1 (1952): 65–98.

Berg, J. Bolzano's Logic. Stockholm: Almqvist and Wiksell, 1962.

Bolzano, Bernard. Grundlegung der Logik. Hamburg, 1964. A useful selection by Friedrich Kambartel from the first two volumes of the Wissenschaftslehre.

Bolzano, Bernard. Wissenschaftslehre, 4 vols. Sulzbach, 1837; Leipzig: Meiner, 1929–1931 (edited by W. Schulz).

Scholz, Heinrich. "Die Wissenschaftslehre Bolzanos." Abhandlungen des Frieśschen Schule, n.s., 6 (1937): 399–472.

Yehoshua Bar-Hillel (1967)

Precursors of Modern Logic: Bolzano

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